Multipolar social systems: Measuring polarization beyond dichotomous contexts

Social polarization is a growing concern worldwide, as it strains social relations, erodes trust in institutions, and thus hurts democratic societies. Polarization has been traditionally studied in binary conflicts where two groups support opposite ideas. However, in many social systems, such as multi-party democracies, political conflicts involve multiple dissenting factions. Despite the prevalence of multipolar systems, there is still a lack of suitable analytical tools to study their polarization patterns. In this work, we introduce new polarization metrics for multidimensional scenarios and develop a methodology that extracts the ideological structure of multipolar contexts from social networks. We propose the trace of the covariance matrix (the total variation) of the multipolar opinion distribution as a measure of global polarization, and its eigendecomposition to identify the directions of maximum polarization. Instead of using a pre-conceived opinion space (conservative vs progressive, liberal vs authoritarian, etc.), our framework reveals the natural ideological axes of the system. We apply our methodology to quadripolar and pentapolar real-world democratic processes, finding non-trivial ideological structures with clear connections to the underlying social context. Our framework opens new avenues for understanding multilateral social tensions and facing the challenges of polarization in a unified way.